A Studentized Spherical Harmonics–Based Nonparametric Two-Sample Test for Compositional and Directional Data

Compositional data analysis has gained increased attention due to the widespread occurrence of simplex-valued data, including microbiome data and financial portfolios. Existing compositional two-sample tests often require $\log$-transformations and only detect mean differences, motivating the need for a more general framework without relying on $\log$-based methods. There is a close connection between compositional data and directional statistics, and we construct a unified non-parametric two-sample test framework. Our work is based on a studentized energy statistic constructed from spherical harmonics theory over a fixed dimensional underlying space, incorporating U-statistics theory and recent developments of studentization for both compositional and directional data. We establish asymptotic normality for our spherical harmonics based test statistics, thus avoiding the need for permutation tests or bootstrap procedures. Our proposed framework sheds new light on the connections between Non-Euclidean data analysis and classical asymptotic high-dimensional data techniques.